Lemma 66.34.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ Let $n \geq 0$. Assume $f$ is locally of finite presentation. The open

of Lemma 66.34.4 is retrocompact in $|X|$. (See Topology, Definition 5.12.1.)

Lemma 66.34.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ Let $n \geq 0$. Assume $f$ is locally of finite presentation. The open

\[ W_ n = \{ x \in |X| \text{ such that the relative dimension of }f\text{ at } x \leq n\} \]

of Lemma 66.34.4 is retrocompact in $|X|$. (See Topology, Definition 5.12.1.)

**Proof.**
Choose a diagram

\[ \xymatrix{ U \ar[r]_ h \ar[d]_ a & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective and étale, see Spaces, Lemma 64.11.6. In the proof of Lemma 66.34.4 we have seen that $a^{-1}(W_ n) = U_ n$ is the corresponding set for the morphism $h$. By Morphisms, Lemma 29.28.6 we see that $U_ n$ is retrocompact in $U$. The lemma follows by definition of the topology on $|X|$, compare with Properties of Spaces, Lemma 65.5.5 and its proof. $\square$

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